GATE 2024 DA – Question 59

Probability and Statistics · Conditional expectation and variance · 2 marks · Numerical answer

Consider a joint probability density function of two random variables $X$ and $Y$:
$f_{X,Y}(x, y) = \begin{cases} 2xy, & 0 < x < 2,\ 0 < y < x \\ 0, & \text{otherwise} \end{cases}$

Then, $E[Y|X = 1.5]$ is ______.

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Correct answer: 1

Explanation

The marginal density of $X$ is $f_X(x) = \int_0^x 2xy\,dy = x^3$. The conditional density is $f(y|x) = \frac{2xy}{x^3} = \frac{2y}{x^2}$ for $0 < y < x$. So $E[Y|X = x] = \int_0^x y\cdot\frac{2y}{x^2}dy = \frac{2}{x^2}\cdot\frac{x^3}{3} = \frac{2x}{3}$. At $x = 1.5$ this is 1.